Polytope of Type {30,4} with group SmallGroup(720,784)

Overview

Group: SmallGroup(720,784)
Rank: 3
Schläfli Type: {30,4}
Vertices, edges, …: 90, 180, 12
Order of s₀s₁s₂: 20
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

5-fold

{6,4}*144

9-fold

{10,4}*80

10-fold

{6,4}*72

18-fold

{10,2}*40

36-fold

{5,2}*20

45-fold

{2,4}*16

90-fold

{2,2}*8

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^3*s2*s1*s2> of order 2

7 facets

2 of {15}*30
5 of {30}*60

45 vertex figures

45 of {4}*8

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s1*s2> of order 2

6 facets

6 of {30}*60

45 vertex figures

45 of {4}*8

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 3

8 facets

6 of {10}*20
2 of {30}*60

30 vertex figures

30 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^2> of order 3

4 facets

4 of {30}*60

30 vertex figures

30 of {4}*8

P/N, where N=<(s0*s1)^3*s0*s2*s1*s0*s1*s2, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1> of order 6

5 facets

2 of {15}*30
3 of {10}*20

15 vertex figures

15 of {4}*8

Representations

Permutation Representation (GAP)

s0 := ( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(16,31)(17,35)(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)(30,37);;
s1 := ( 1,17)( 2,16)( 3,20)( 4,19)( 5,18)( 6, 7)( 8,10)(11,42)(12,41)(13,45)(14,44)(15,43)(21,37)(22,36)(23,40)(24,39)(25,38)(26,27)(28,30)(31,32)(33,35);;
s2 := (16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(45)!( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(16,31)(17,35)(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)(30,37);
s1 := Sym(45)!( 1,17)( 2,16)( 3,20)( 4,19)( 5,18)( 6, 7)( 8,10)(11,42)(12,41)(13,45)(14,44)(15,43)(21,37)(22,36)(23,40)(24,39)(25,38)(26,27)(28,30)(31,32)(33,35);
s2 := Sym(45)!(16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40);
poly := sub<Sym(45)|s0,s1,s2>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 >;

References

None.

to this polytope.